cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A139292 2^(2p - 1)/8, where p is prime.

Original entry on oeis.org

1, 4, 64, 1024, 262144, 4194304, 1073741824, 17179869184, 4398046511104, 18014398509481984, 288230376151711744, 1180591620717411303424, 302231454903657293676544, 4835703278458516698824704
Offset: 1

Views

Author

Omar E. Pol, Apr 13 2008

Keywords

Crossrefs

Cf. A139286, A139287, A139288, A139289, A139290, A139291, A139293, A139294, A139306.

Programs

  • Mathematica
    2^(2#-1)/8&/@Prime[Range[20]] (* Harvey P. Dale, Aug 19 2012 *)

Formula

a(n) = A139286(n)/8.

Extensions

More terms from Jon E. Schoenfield, Jul 16 2010

A139293 (2^(2p - 1)/8)-1, where p is prime.

Original entry on oeis.org

0, 3, 63, 1023, 262143, 4194303, 1073741823, 17179869183, 4398046511103, 18014398509481983, 288230376151711743, 1180591620717411303423, 302231454903657293676543, 4835703278458516698824703
Offset: 1

Views

Author

Omar E. Pol, Apr 13 2008

Keywords

Crossrefs

Cf. A139286, A139287, A139288, A139289, A139290, A139291, A139292, A139294, A139306.

Programs

  • Mathematica
    (2^(2#-1))/8-1&/@Prime[Range[30]] (* Harvey P. Dale, May 09 2012 *)

Formula

a(n) = (A139286(n)/8)-1.

Extensions

More terms from Jon E. Schoenfield, Jul 16 2010

A139116 a(n) = p*(p-1)/2, where p is A000043(n).

Original entry on oeis.org

1, 3, 10, 21, 78, 136, 171, 465, 1830, 3916, 5671, 8001, 135460, 183921, 817281, 2425503, 2600340, 5172936, 9041878, 9779253, 46933516, 49406770, 62860078, 198732016, 235455850, 269317236, 989969256, 3718884403, 6105401253, 8718403176, 23347552095, 286402257541
Offset: 1

Views

Author

Omar E. Pol, May 10 2008

Keywords

Links

Crossrefs

Cf. A000043, A000217, A036689, A139306, A139307, A139116, A139223, A139224, A139225, A139226.

Programs

  • Mathematica
    (#(#-1))/2&/@MersennePrimeExponent[Range[47]] (* Harvey P. Dale, Aug 13 2021 *)

Formula

a(n) = A000043(n)*(A000043(n)-1)/2.

Extensions

a(24)-a(32) from Harvey P. Dale, Aug 13 2021

A139291 a(n) = 2^(2*prime(n) - 3) - 1.

Original entry on oeis.org

1, 7, 127, 2047, 524287, 8388607, 2147483647, 34359738367, 8796093022207, 36028797018963967, 576460752303423487, 2361183241434822606847, 604462909807314587353087, 9671406556917033397649407, 2475880078570760549798248447, 10141204801825835211973625643007
Offset: 1

Views

Author

Omar E. Pol, Apr 13 2008

Keywords

Crossrefs

Cf. A139286, A139287, A139288, A139289, A139290, A139293, A139294, A139306.

Programs

  • Mathematica
    (2^(2#-1))/4-1&/@Prime[Range[20]] (* Harvey P. Dale, May 08 2011 *)

Formula

a(n) = A139286(n)/4 - 1.
a(n) = A000225(A131426(n)). - Max Alekseyev, Mar 07 2020
a(n) = A139290(n) - 1. - Omar E. Pol, Mar 10 2020

Extensions

Edited by Max Alekseyev, May 03 2009, Mar 07 2020

A139115 a(n) = p*(p - 1), where p is A000043(n).

Original entry on oeis.org

2, 6, 20, 42, 156, 272, 342, 930, 3660, 7832, 11342, 16002, 270920, 367842, 1634562, 4851006, 5200680, 10345872, 18083756, 19558506, 93867032, 98813540, 125720156, 397464032, 470911700, 538634472, 1979938512, 7437768806, 12210802506
Offset: 1

Views

Author

Omar E. Pol, May 10 2008

Keywords

Links

Crossrefs

Cf. A000043, A036689, A139306, A139307, A139116, A139223, A139224, A139225, A139226.

Programs

  • Mathematica
    #(#-1)&/@MersennePrimeExponent[Range[30]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jan 15 2020 *)

Formula

a(n) = A000043(n)*(A000043(n) - 1).

Extensions

More terms from Vincenzo Librandi, May 11 2010

A139226 M(M-1)/6, where M is Mersenne prime A000668(n).

Original entry on oeis.org

1, 7, 155, 2667, 11180715, 2863245995, 45812722347, 768614335330822827, 886151997189943914116283202246716075, 63853980869412035764931125821777872829435728032869035, 4388012152856549445746584486819520216982214368341183846591146667, 4824670384888174809315457708695329493801178769338122219111555681805105015467
Offset: 1

Views

Author

Omar E. Pol, May 10 2008

Keywords

Comments

Perfect number A000396(n) minus Mersenne prime A000668(n), divided by 3.
Terms from a(13) on have 313 or more digits and are not listed for that reason. - R. J. Mathar, May 11 2008

Links

Crossrefs

Cf. A000217, A000396, A000668, A036689, A139096, A139115, A139116, A139223, A139224, A139225, A139256, A139306.

Formula

a(n) = A000668(n)*(A000668(n)-1)/6 = A139223(n)/6 = A139224(n)/3.
a(n) = (A000396(n)-A000668(n))/3.

Extensions

More terms from R. J. Mathar, May 11 2008

A139307 a(n) = (2^(2*p - 1)) - 1, where p is A000043(n).

Original entry on oeis.org

7, 31, 511, 8191, 33554431, 8589934591, 137438953471, 2305843009213693951, 2658455991569831745807614120560689151, 191561942608236107294793378393788647952342390272950271
Offset: 1

Views

Author

Omar E. Pol, Apr 13 2008, May 08 2008

Keywords

Comments

Ultraperfect numbers (A139306) minus 1.

Examples

			a(5) = 33554431 because A000043(5) = 13 and (2^(2*13 - 1))-1 = 2^25 - 1 = 33554431.

Links

Crossrefs

Cf. A000043, A139287, A139295, A139306.

Programs

  • Mathematica
    2^(2 * MersennePrimeExponent[Range[10]] - 1) - 1 (* Amiram Eldar, Oct 17 2024 *)

Formula

a(n) = (2^(2*A000043(n) - 1)) - 1 = A139306(n) - 1.

A152921 a(n) = 2^(2p-1)/2, where p is A000043(n).

Original entry on oeis.org

4, 16, 256, 4096, 16777216, 4294967296, 68719476736, 1152921504606846976, 1329227995784915872903807060280344576, 95780971304118053647396689196894323976171195136475136, 6582018229284824168619876730229402019930943462534319453394436096
Offset: 1

Views

Author

Omar E. Pol, Dec 15 2008

Keywords

Comments

Ultraperfect numbers (A139306), divided by 2.
Also, a(n) is the largest proper divisor of the n-th ultraperfect number.
The cototient (A051953) of the even perfect numbers (A000396). - Amiram Eldar, Mar 06 2022
These cototients are squares = (2^(p-1))^2. - Bernard Schott, Mar 14 2022

Crossrefs

Cf. A000043, A000396, A019279, A051953, A061652, A139288, A139306, A152922, A152923.

Programs

  • Mathematica
    a[n_] := 4^(MersennePrimeExponent[n] - 1); Array[a, 12] (* Amiram Eldar, Mar 06 2022 *)

Formula

a(n) = A139306(n)/2.
a(n) = A051953(A000396(n)), if there are no odd perfect numbers. - Amiram Eldar, Mar 06 2022
a(n) = A061652(n)^2. - Bernard Schott, Mar 14 2022

Extensions

More terms from Amiram Eldar, Mar 06 2022

A152922 a(n) = 2^(2*p-1)/4, where p is A000043(n).

Original entry on oeis.org

2, 8, 128, 2048, 8388608, 2147483648, 34359738368, 576460752303423488, 664613997892457936451903530140172288, 47890485652059026823698344598447161988085597568237568
Offset: 1

Views

Author

Omar E. Pol, Dec 15 2008

Keywords

Comments

Ultraperfect numbers (A139306), divided by 4.

Links

Crossrefs

Cf. A000043, A139290, A139306, A152921, A152923.

Programs

  • Mathematica
    2^(2 * MersennePrimeExponent[Range[10]] - 3) (* Amiram Eldar, Oct 17 2024 *)

Formula

a(n) = A139306(n)/4 = A152921(n)/2.

Extensions

a(9)-a(10) from Amiram Eldar, Oct 17 2024

A139096 Infraperfect numbers: a(n) = 2^(2*p - 1) - 2^p, where p is A000043(n).

Original entry on oeis.org

4, 24, 480, 8064, 33546240, 8589803520, 137438429184, 2305843007066210304, 2658455991569831743501771111346995200, 191561942608236107294793377774818628309652252823388160
Offset: 1

Views

Author

Omar E. Pol, Apr 22 2008

Keywords

Comments

Difference between n-th even perfect number and n-th even superperfect number A061652(n). Difference between n-th ultraperfect number A139306(n) and n-th Mersenne prime A000668(n), minus 1. Also, difference between n-th perfect number A000396(n) and n-th superperfect number A019279(n), if there are no odd perfect and superperfect numbers.

Examples

			a(2) = 24 because A000043(2) = 3 then 2^(2*3 - 1) - 2^3 = 2^5 - 2^3 = 32 - 8 = 24.

Links

Crossrefs

Cf. A000043, A000396, A000668, A019279, A061652, A072868, A139306.

Programs

  • Mathematica
    Map[2^(2*#-1) - 2^# &, MersennePrimeExponent[Range[10]]] (* Amiram Eldar, Oct 17 2024 *)

Formula

a(n) = 2^(2*A000043(n) - 1) - 2^A000043(n) = A139306(n) - 2^A000043(n) = A139306(n) - A000668(n) - 1 = A139306(n) - (A000668(n)+1) = A139306(n) - 2*A061652(n) = A139306(n) - A072868(n).

Extensions

More terms from R. J. Mathar, Feb 05 2010
Previous Showing 21-30 of 33 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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